[Paper Review] A Compactness Theorem for Riemannian Manifolds with Boundary and Applications

Kenneth S. Knox|arXiv (Cornell University)|Nov 27, 2012

Geometric Analysis and Curvature Flows4 citations

TL;DR

This paper establishes a weak $L^{1,p}$ and H"older ($C^{ au}$) compactness theorem for uniformly mean-convex Riemannian 3-manifolds with boundary under bounds on curvature, diameter, boundary area, and mean curvature. The key result enables two geometric stability theorems: manifolds with Ricci curvature near zero and boundary close to a round sphere are $C^{ au}$-close to a Euclidean ball, and those with Ricci curvature near $\pm1$ and mean curvature near $2$, $2\sqrt{2}$, or $0$ are $C^\tau$-close to metric balls in space forms of curvature $0$, $-1$, or $1$, respectively.

ABSTRACT

In this paper we prove weak L^{1,p} (and thus C^α) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We obtain two stability theorems from the compactness result. The first theorem applies to 3-manifolds (contained in the aforementioned class) that have Ricci curvature close to 0 and whose boundaries are Gromov-Hausdorff close to a fixed metric on S^2 with positive curvature. Such manifolds are C^α close to the region enclosed by a Weyl embedding of the fixed metric into \R^3. The second theorem shows that a 3-manifold with Ricci curvature close to 0 (resp. -2, 2) and mean curvature close to 2 (resp. 2\sqrt 2, 0) is C^α close to a metric ball in the space form of constant curvature 0 (resp -1, 1), provided that the boundary is a topological sphere.

Motivation & Objective

• To establish a compactness theorem for Riemannian $n$-manifolds with boundary under curvature, diameter, boundary area, and mean curvature bounds.
• To extend Cheeger-Gromov and Kodani-type compactness results to include mean-convexity and pointwise mean curvature control.
• To derive geometric stability theorems for 3-manifolds with boundary under Ricci curvature and mean curvature constraints.
• To show that sequences of such manifolds converge in $C^{\tau}$ and weak $L^{1,p}$ topology to limits with controlled curvature and boundary geometry.

Proposed method

• Prove weak $L^{1,p}$ and $C^{\tau}$ compactness for the class of uniformly mean-convex Riemannian $n$-manifolds with boundary under curvature, diameter, boundary area, and mean curvature bounds.
• Use harmonic coordinates and intrinsic analysis to obtain $L^{2,p}$ control of the metric in the interior and on the boundary.
• Control the normal component of the metric near the boundary via pointwise bounds on mean curvature, yielding $L^{1,p}$ regularity.
• Apply elliptic regularity and the theory of holomorphic quadratic differentials to show that constant mean curvature immersions of $S^2$ into $\mathbb{R}^3$ or $\mathbb{H}^3$ must be spheres or equators.
• Use the developing map and convergence in $L^{1,p}$ to show that the limit metric has constant curvature and is isometric to a space form ball.
• Leverage Almgren's result on minimal immersions and holomorphic quadratic differentials to prove that $C^{2,\epsilon}$ isometric immersions of $S^2$ into $S^3$ with constant second fundamental form must be equators.

Experimental results

Research questions

• RQ1 Can a compactness theorem be established for Riemannian manifolds with boundary under curvature, diameter, boundary area, and mean curvature bounds, even with only pointwise control on mean curvature?
• RQ2 Under what conditions does a sequence of 3-manifolds with Ricci curvature close to zero and boundary close to a round sphere converge to a Euclidean ball in $\mathbb{R}^3$?
• RQ3 When does a 3-manifold with Ricci curvature near $-1$, $0$, or $1$ and mean curvature near $2\sqrt{2}$, $2$, or $0$ converge to a metric ball in the corresponding space form?
• RQ4 To what extent does the $C^{\tau}$ regularity of the limit metric follow from weak $L^{1,p}$ convergence and curvature bounds?
• RQ5 Can the holomorphic quadratic differential argument be extended from analytic to $C^{2,\epsilon}$ isometric immersions to prove rigidity of constant mean curvature surfaces?

Key findings

• The class of uniformly mean-convex Riemannian $n$-manifolds with boundary satisfying $|\sec(M)|\leq K$, $\textrm{diam}(M)\leq D$, $\operatorname{area}(\partial M)\geq A_0$, and $0 < 1/H_0 < H < H_0$ is precompact in the $C^{\tau}$ and weak $L^{1,p}$ topologies for any $0<\tau<1$ and $p<\infty$.
• If a sequence of 3-manifolds has Ricci curvature $\to 0$, mean curvature $\to 2$, and boundary Gromov-Hausdorff close to $S^2$ with positive curvature, then the limit is $C^{\tau}$-close to a Euclidean ball.
• If a 3-manifold has $\chi(\partial M)=2$, Ricci curvature $\to -2$, and mean curvature $\to 2\sqrt{2}$, then it is $C^{\tau}$-close to a metric ball in $\mathbb{H}^3$ of curvature $-1$.
• If a 3-manifold has $\chi(\partial M)=2$, Ricci curvature $\to 2$, and mean curvature $\to 0$, then it is $C^{\tau}$-close to a metric ball in $\mathbb{S}^3$ of curvature $1$.
• The developing map of the limit metric induces a $C^{2,\epsilon}$ isometric immersion of $S^2$ into $\mathbb{R}^3$ or $\mathbb{H}^3$, and such immersions with constant second fundamental form are necessarily spheres or equators.
• Using holomorphic quadratic differentials and elliptic regularity, the limit metric is shown to be $C^{\infty}$ and isometric to a ball in a space form of constant curvature $0$, $-1$, or $1$.